Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C4×#3417 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,3,4);(5,6,11,16)(7,17,21,45)(8,13,22,36)(9,25,29,43)(10,30,27,15)(12,32,35,19)(14,39,41,28)(18,44,46,66)(20,49,31,40)(23,52,50,65)(24,55,51,38)(26,42,57,34)(33,58,59,47)(37,62,61,54)(48,67,56,63)(53,64,68,60);(5,7,11,21)(6,12,16,35)(8,18,22,46)(9,26,29,57)(10,31,27,20)(13,33,36,59)(14,40,41,49)(15,42,30,34)(17,43,45,25)(19,39,32,28)(23,53,50,68)(24,56,51,48)(37,63,61,67)(38,64,55,60)(44,65,66,52)(47,62,58,54);(5,8)(6,13)(7,18)(9,23)(10,24)(11,22)(12,33)(14,37)(15,38)(16,36)(17,44)(19,47)(20,48)(21,46)(25,52)(26,53)(27,51)(28,54)(29,50)(30,55)(31,56)(32,58)(34,60)(35,59)(39,62)(40,63)(41,61)(42,64)(43,65)(45,66)(49,67)(57,68);(8,22)(13,36)(18,46)(23,50)(24,51)(33,59)(37,61)(38,55)(44,66)(47,58)(48,56)(52,65)(53,68)(54,62)(60,64)(63,67)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529813);